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A085104 Primes of the form 1 + n + n^2 + n^3 + ... + n^k, n > 1, k > 1. 37
7, 13, 31, 43, 73, 127, 157, 211, 241, 307, 421, 463, 601, 757, 1093, 1123, 1483, 1723, 2551, 2801, 2971, 3307, 3541, 3907, 4423, 4831, 5113, 5701, 6007, 6163, 6481, 8011, 8191, 9901, 10303, 11131, 12211, 12433, 13807, 14281, 17293, 19183, 19531, 20023 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Primes that are base-b repunits with three or more digits for at least one b >= 2: Primes in A053696. Subsequence of A000668 UNION A076481 UNION A086122 UNION A165210 UNION A102170 UNION A004022 UNION ... (for each possible b). - Rick L. Shepherd, Sep 07 2009

Also known as Brazilian primes.  The primes that are not Brazilian primes are in A220627. - Bernard Schott, Dec 18 2012

The number of terms k+1 is always an odd prime, but this is not enough to guarantee a prime, for example 111 = 1 + 10 + 100 = 3*37. - Bernard Schott, Dec 18 2012

The inverses of the Brazilian primes form a convergent series, and the sum is between 0.33 and 1. (See Theorem 4 of Quadrature article in the Links.) - Bernard Schott, Dec 18 2012

It is not known whether there are infinitely many Brazilian primes. See A002383. - Bernard Schott, Jan 11 2013

Primes of the form (n^p - 1)/(n - 1), where p is odd prime and n > 1. - Thomas Ordowski, Apr 25 2013

Number of terms less than 10^n: 1, 5, 14, 34, 83, 205, 542, 1445, 3880, 10831, 30699, 88285, ..., . - Robert G. Wilson v, Mar 31 2014

From Bernard Schott, Apr 08 2017: (Start)

Brazilian primes fall into two classes:

1) when n is prime, we get sequence A023195 except 3 which is not Brazilian,

2) when n is composite, we get sequence A285017. (End)

LINKS

T. D. Noe, Table of n, a(n) for n = 1..3880 (primes < 10^9)

Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avril-juin 2010, pages 30-38; included here with permission from the editors of Quadrature.

FORMULA

A010051(a(n)) * A088323(a(n)) > 1. - Reinhard Zumkeller, Jan 22 2014

EXAMPLE

13 is a term since it is prime and 13 = 1 + 3 + 3^2 = (111)_3.

31 is a term since it is prime and 31 = 1 + 2 + 2^2 + 2^3 + 2^4 = (11111)_2.

From Hartmut F. W. Hoft, May 08 2017 :(Start)

The sequence represented as a sparse matrix with the k-th column indexed by A006093(k+1), primes minus 1, and row n by A000027(n+1). Traversing the matrix by counterdiagonals produces a non-monotone ordering.

    2    4      6        10             12          16

2  7    31     127      -              8191        131071

3  13   -      1093     -              797161      -

4  -    -      -        -              -           -

5  31   -      19531    12207031       305175781   -

6  43   -      55987    -              -           -

7  -    2801   -        -              16148168401 -

8  73   -      -        -              -           -

9  -    -      -        -              -           -

10  -    -      -        -              -           -

11  -    -      -        -              -           50544702849929377

12  157  22621  -        -              -           -

13  -    30941  5229043  -              -           -

14  211  -      8108731  -              -           -

15  241  -      -        -              -           -

16 -    -      -        -              -           -

17  307  88741  25646167 2141993519227  -           -

18  -    -      -        -              -           -

19  -    -      -        -              -           -

20  421  -      -        10778947368421 -           689852631578947368421

21  463  -      -        17513875027111 -           1502097124754084594737

22  -    245411 -        -              -           -

23  -    292561 -        -              -           -

24  601  346201 -        -              -           -

Except for the initial values in the respective sequences the rows and columns as labeled in the matrix are:

column  2:  A002383            row 2:  A000668

column  4:  A088548            row 3:  A076481

column  6:  A088550            row 4:  -

column 10:  A162861            row 5:  A086122.

(End)

MATHEMATICA

max = 140; maxdata = (1 - max^3)/(1 - max); a = {}; Do[i = 1; While[i = i + 2; cc = (1 - m^i)/(1 - m); cc <= maxdata, If[PrimeQ[cc], a = Append[a, cc]]], {m, 2, max}]; Union[a] (* Lei Zhou, Feb 08 2012 *)

f[n_] := Block[{i = 1, d, p = Prime@ n}, d = Rest@ Divisors[p - 1]; While[ id = IntegerDigits[p, d[[i]]]; id != Reverse@ id || Union@ id != {1}, i++]; d[[i]]]; Select[ Range[2, 60], 1 + f@# != Prime@# &] (* Robert G. Wilson v, Mar 31 2014 *)

PROG

(PARI) list(lim)=my(v=List(), t, k); for(n=2, sqrt(lim), t=1+n; k=1; while((t+=n^k++)<=lim, if(isprime(t), listput(v, t)))); vecsort(Vec(v), , 8) \\ Charles R Greathouse IV, Jan 08 2013

(Haskell)

a085104 n = a085104_list !! (n-1)

a085104_list = filter ((> 1) . a088323) a000040_list

-- Reinhard Zumkeller, Jan 22 2014

(PARI) A085104_vec(n)={my(h=vector(n, i, 1), y, c, z=4, L:list); L=List(); forprime(x=3, , for(m=z, x-1, y=digits(x, m); if((y==h[1..#y])&&2<#y, listput(L, x); z=m; if(c++==n, return(Vec(L))))))} \\ R. J. Cano, Apr 18 2017

CROSSREFS

Cf. A002383, A189891 (complement), A125134.

Cf. A003424 (n restricted to prime powers), A059055.

Cf. A053696, A000668, A076481, A086122, A165210, A102170, A004022, A220627, A086930.

Equals A023195 \3 Union A285017, but A023195 Intersection A285017 = Empty. - Bernard Schott, Apr 08 2017

Sequence in context: A231626 A110912 A240680 * A162652 A181141 A031158

Adjacent sequences:  A085101 A085102 A085103 * A085105 A085106 A085107

KEYWORD

nonn

AUTHOR

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jul 03 2003

EXTENSIONS

More terms from David Wasserman, Jan 26 2005

STATUS

approved

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Last modified June 22 23:15 EDT 2017. Contains 288633 sequences.